### Generalized local homology and local cohomology modules

Time: 14:00 to  15:30 Ngày 26/03/2019

Venue/Location: C2-714, VIASM

Speaker: PGS. TS. Trần Tuấn Nam

Content:
It is well-known that the local cohomology theory introduced by Grothendieck in 1967 has proved to be an important tool in algebraic geometry and commutative algebra.
Duality of the local cohomology theory is the theory of local homology which also plays an important role. Let \$I\$ be an ideal of a netherian commutative ring  \$R\$ and \$M\$ an \$R-\$module, the \$i-\$th local cohomology module \$H_I^i(M)\$ and the \$i-\$th local homology module \$H^I_i(M)\$ of \$M\$ with respect to \$I\$ are defined by \$H_I^i(M) = \underset{t}\varinjlim {\Ext}_R^i(R/I^t,M).\$ and \$H^I_i(M) = \underset{t}\varprojlim {\Tor}^R_i(R/I^t,M).\$
Then, many mathematicians have sought to extend the concept of local cohomology and obtained many interesting results. In 1970 Herzog  introduced the concept of generalized local cohomology. In 2007 Peter Schenzel introduced the concept of formal local cohomology. In 2008 Ryo Takahashi, Yuji Yoshino and Takeshi Yoshizawa introduced the concept of local cohomology with respect to a pair of ideals (I, J). Similarly, we also have corresponding extensions of the concept of local homology modules.
In this report, we will present some important properties of these modules.